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The chain rule is most commonly seen in Leibniz's notation, which is as follows:
dz/dx = dz/dy * dy/dx
You can remember it intuitively by thinking of the two 'dy' terms cancelling to leave dz/dx.
So why use the chain rule?
You are used to differentiating equations in the form y = f(x), but say both sides of the equation where functions eg g(y) = f(x) and you had to differentiate the equation with respect to x.
g is a function of y, not x, so you can't simply calculate dg(y)/dx like you can df(x)/dx. Using the chain rule we can express dg(y)/dx as dg(y)/dy * dy/dx. These two terms can be calculated (assuming y is a function of x). This is really what the chain rule is saying: that the derivative of a function composition can be expressed as a product of the respective derivatives.
Another example of when the chain rule might come in useful is in mechanics: Acceleration is defined as the derivative of velocity: dv/dt. Sometimes though it might be useful to integrate acceleration of a distance, x, rather than over time. To eliminate time from this expression we can use the chain rule by saying dv/dt = dv/dx * dx/dt. Then noting that dx/dt is in fact velocity (v = dx/dt) we can write that dv/dt = v * dv/dx thus making acceleration a function only of velocity and position.see more